Two functional inequalities

\textbf{Problem 1.} Find all functions f:\mathbb{R}\to\mathbb{R} such that for all x and y holds

f(x)f(x+y)\ge f(x)^2+xy.


\textbf{Problem 2.} Find all differentiable functions f:\mathbb{R}\to\mathbb{R} with f(0)=0,\ f(1)=1 and such that for all x and y holds

f(x+y)\ge 2022^xf(x)+f(y).

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